sign in sign up
<<
Home , Formation rule, Lambda calculus, Montague grammar

Formal , Lecture 2 Formal Semantics, Lecture 2 B. Partee, MGU, February 22, 2005 p.1 B. Partee, MGU, February 22, 2005 p.2

Lecture 2. Lambda abstraction, NP semantics, and a Fragment of English

(b) NP 1. Lexical and Structural ...... 1 | 2. Lambdas ...... 3 CNP 2.1. A first-order part of the lambda-...... 3 3 2.2. The typed ...... 4 ADJ CNP 3. Montague’s semantics for Noun Phrases...... 5 | 3.1. Semantics via direct model-theoretic of English...... 5 9 3.2. Semantics via translation from English into a logical language...... 5 old CNP and CNP 4. English Fragment 1...... 5 | | 4.0 Introduction ...... 5 men women 4.1. Syntactic categories and their semantic types...... 6 4.2. Syntactic Rules and Semantic Rules...... 6 (2) Every student read a book. ( ambiguity) 4.2.1. Basic syntactic rules...... 7 4.2.2. Semantic interpretation of the basic rules...... 7 Just one (surface) syntactic structure: 4.2.3. Rules of Relative clauses, Quantification, Phrasal . See Section 6...... 8 4.2.4. Type multiplicity and type shifting...... 8 S 4.3. Lexicon...... 9 3 5. Examples ...... 11 NP VP 6. Rules of Relative clause formation, Quantifying In, Phrasal Negation...... 11

6.1. (Restrictive) Relative clause formation...... 11 3 3 6.2. Quantifying In...... 12 DET CNP V NP 6.3. Conjunction...... 13 | | | 3 6.4. Phrasal and lexical negation...... 14 every student read DET CNP Appendix: Montague’s intensionsal , with lambdas and types...... 15 | | A.1 Introduction...... 15 a book A.2. (IL)...... 15 A.2.1. Types and model structures...... 15 logic representations of the two readings: A.2.2. Atomic expressions (“lexicon”), notation, and interpretation...... 16 (i) ∀x ( Student (x) → y ( Book (y) & Read (x,y)) A.2.3. Syntactic rules and their model-theoretic semantic interpretation ...... 17 › ...... 18 (ii) ∃y ( Book (y) & ∀x ( Student (x) → Read (x,y)) HOMEWORK #1, Due March 8...... 19 HELP FOR HOMEWORK #1...... 21 Compositional interpretation of the English sentence: ??. More below.

NOTE: I know this is too much for one lecture. But it is useful to have all of this in one handout. We will spend this time and part of next time on this, with more next time about the semantics of noun phrases as Generalized The difficulty for compositionality if we try to use predicate calculus to represent “logical Quantifiers. form”: What is the interpretation of “every student”? There is no appropriate syntactic category or semantic type in predicate logic. Inadequacy of 1st-order predicate logic for 1. Lexical and Structural Ambiguity representing the semantic structure of . We can solve this problem when we

Lexical ambiguity: bank1, bank2 : both CN (common noun), homonyms; have the lambda-calculus and a richer . open1 (ADJ), open2 (IV) (intransitive verb), open3 (TV) (transitive verb). Categories of PC: Categories of NL: Structural ambiguity. Compositionality requires a “disambiguated language” (a “language Formula - Sentence without ambiguity”). So we interpret expressions with syntactic structure, not just strings. Predicate - Verb, Common Noun, Adjective (1) old men and women. Two meanings, two structures. “old” applies only to “men”, or to Constant - Proper Noun “men and women”. - Pronoun (he, she, it)

======(a) NP (no more) - Verb Phrase, Noun Phrase, Common Noun Phrase, Adjective 9 NP and NP Phrase, Determiner, Preposition, Prepositional Phrase, Adverb, | | CNP CNP 3 | ADJ CNP women | | old men MGU052.doc 02/21/05 1:15 AM MGU052.doc 02/21/05 1:15 AM

Formal Semantics, Lecture 2 Formal Semantics, Lecture 2 B. Partee, MGU, February 22, 2005 p.3 B. Partee, MGU, February 22, 2005 p.4

2. Lambdas Rule for combining CNP and REL: λy[CNP’(y) & REL’(y)] (combining “translations”)

2.1. A first-order part of the lambda-calculus. Compositional translation of the syntactic structure above into λ-calculus: (read bottom-to- To begin looking at the lambda calculus, we will start with just a “first-order” part of top) it, as if we were just adding a bit of the lambda calculus to the predicate calculus rules from λy[man(y) & λz[love (z,m)] (y)] Lecture 1. Then in section 2.2. we will look at the fully as given in 3 Montague’s Intensional Logic Rule 7 in the Appendix. man λz[love (z,m)] | | Lambda-abstraction rule, first version. man love (z, m) λ-abstraction applies to formulas to make predicates. This extends PC in a way that allows us to represent more complex Common Noun Phrases, Adjective Phrases, some Verb By λ-conversion (see below), the top line is equivalent to: λy[man(y) & love (y,m)] Phrases. For some other categories we will need the full version of the λ-abstraction rule. R9: If ϕ ∈ Form and v is a variable, then λv[ϕ] ∈ Pred-1. 2.2. The typed lambda calculus. S9: ›λv[ϕ]M, g is the S of all d 0 D such that || ϕ ||M, g [d/v] = 1. Examples. For all examples, assume that we start with an assignment g such that g(v) = John The full version of the typed lambda calculus fits into Montague’s intensional logic with its for all v. In most of the examples below, the choice of initial assignment makes no type theory; see the Appendix for a complete statement of Montague’s intensional logic. The difference. And assume that I(b) = Bill, I(m) = Mary. parts we will use the most will be the type theory, the lambda calculus (Rule 7), and the rule of “functional application” (Rule 6). Montague’s intensional logic includes the predicate M, g (i) ›|| λx[run(x)] || = the set of all individuals that run. calculus as a subpart (see Rule 2), but not restricted to first-order: we can quantify over variables of any type. (ii) ›|| λx[love(x, b)] ||M, g = the set of all individuals that love || b ||M, g[d/x], i.e. I(b), i.e. Bill.

M, g M, g[d/x] (iii) ›|| λx[love(x, y)] || = the set of all individuals that love || y || , i.e. g(y), i.e. John. Lambda-abstraction, full version.

(iv) ›|| λx[fish (x) & love(x, b)] ||M, g = the set of all fish that love Bill. In general: λ-expressions denote functions. λv[α] denotes a whose is represented by the variable v and whose value (v) to represent “walks and talks”: λy[(walk (y) & talk(y))] for any given value of v is specified by the α.

(vi) to represent “Mary walks and talks” with constituents that correspond to surface : Example: λx[x2 + 1] denotes the function x → x2 + 1. λy[(walk (y) & talk(y))] (m) 2 Function-argument application: λx[x + 1] (5) = 26 Syntactic Structure of the formula in (vi): λ-expressions provide explicit specification of the functions they name, unlike arbitrary Form names like f, g. (The λ-calculus was invented by the logician . The 3 LISP, invented by John McCarthy, was modelled on the λ-calculus.) Pred-1 Term | Syntactic and Semantic Rule: (a restatement of Syntactic and Semantic Rules 7 of IL) 3 α λy Form m R7’: If " is an expression of any type a and v is a variable of type b, then λv[ ] is an → 9 expression of type b a (the type of functions from b-type things to a-type things.) α M, g → Form & Form S7’: || λv[ ]|| is that function f of type b a such that for any object d of type b, M, g[d/v] f(d) = || α || . By λ-conversion (see below), the formula in (vi) is equivalent to (walk(m) & talk(m)). Lambda-conversion: A principle concerning the application of λ-expressions to . (vii) to represent the CNP “man who loves Mary”: Syntactic structure: Examples: λx[x2 + 1] (5) = 52 + 1 = 26 λx[run(x)](b) ≡ run(b) CNP λy[(walk(y) & talk(y))] (m) ≡ (walk(m) & talk(m)) 3 λz[love (z,m)] (y) ≡ love (y,m) CNP REL: who loves Mary | | Lambda-conversion Rule: λv[α](β) ≡ α′ , where α′ is like α but with every free man S: z loves Mary occurrence of v replaced by β.

(Note: Occurrences of v that are free in α are bound by λv in λv[α].) MGU052.doc 02/21/05 1:15 AM MGU052.doc 02/21/05 1:15 AM

Formal Semantics, Lecture 2 Formal Semantics, Lecture 2 B. Partee, MGU, February 22, 2005 p.5 B. Partee, MGU, February 22, 2005 p.6

3. Montague’s semantics for Noun Phrases. 4.1. Syntactic categories and their semantic types. 3.1. Semantics via direct model-theoretic interpretation of English. [See Larson’s chapter 12.] Syntactic Semantic type Expressions category (extensionalized) 3.2. Semantics via translation from English into a logical language. ======ProperN e names (John) S t sentences Indicated by translations of English expressions into the λ-calculus. P is a variable ranging → over sets, i.e. a predicate variable. (In full Montague with intensionality, the CN(P) e t common noun phrases (cat) NP (i) e “e-type” or “referential” NPs (John, the king) analysis uses variables over properties, so sometimes in discussion in , we talk as though (ii)(e→ t) → t noun phrases as generalized quantifiers P were a variable over properties.) (every man, the king, a man, John) (iii) e → t NPs as predicates (a man, the king) John λP[P(j)] ADJ(P) (i) e → t predicative adjectives (carnivorous, happy) John walks λP[P(j)] (walk) ≡ walk (j) (ii)(e → t) → (e → t) adjectives as predicate modifiers (skillful) every student λP[∀x ( student(x) → P(x) )] REL e → t relative clauses (who(m) Mary loves) every student walks λP[∀x ( student(x) → P(x) )] (walk) ≡ ∀x( student(x) → walk(x) ) VP, IV e → t verb phrases, intransitive verbs (loves Mary, is tall, a student λP[∃x ( student(x) & P(x) )] walks) → the king λP[∃x ( king(x) & ∀y ( king(y) → y = x) & P(x) )] TV(P) type(NP) type(VP) transitive verb (phrase) (loves) is (e → t) → (e → t) is (the set of properties which the one and only king has) → DET type(CN) type(NP) a, some, the, every, no 4.2. Syntactic Rules and Semantic Rules. 4. English Fragment 1.

4.0 Introduction Two different approaches to semantic interpretation of natural language syntax (both compositional, both formalized, and illustrated, by Montague): In this part we present a small sample English grammar (a “fragment”, in MG terminology), that is, an explicit description of the syntax and semantics of a small part of English. This A. Direct Model-theoretic interpretation: Semantic values of natural language expressions fragment is intended to serve several purposes: making certain aspects of formal semantics (or their “underlying structure” counterparts) are given directly in model-theoretic terms; no more explicit, including (and illustrating) more of the basics of the lambda-calculus. The intermediate language like Montague’s intensional logic (but for some linguists there is a fragment is of interest in its own right and will also serve as background for the next lecture. syntactic level of “logical form” to which this model-theoretic interpretation applies, so the The fragment, with its very minimal lexicon, also illustrates the typically minimal treatment distinction between the two strategies is not always sharp.) This is the direct “English as a of the lexicon in classical . ” strategy. For illustration, see Heim and Kratzer (1998). Also see the

discussion in Larson’s chapter 12. The semantics of the fragment will be given via translation into Montague’s Intensional

Logic (IL) (the alternatives would be to give a direct model-theoretic interpretation, or an B. Interpretation via translation: Stage 1: compositional translation from natural language interpretation via translation into some other model-theoretically interpreted intermediate to a language of semantic representation, such as Montague’s intensional logic. For an language). In the first part of this lecture we presented Montague’s IL: Its type structure and expression  of category formed from expressions  of category A and  of category B, the model structures in which it is interpreted, and its syntax and model-theoretic semantics. γ α β determine TR( ) as a function of TR( ) and TR( ). Stage 2: the compositional model- Now we introduce the fragment of English: first the syntactic categories and the category- theoretic interpretation rules to the intermediate language. type correspondence, then the basic syntactic rules and the principles of semantic interpretation, and then a small lexicon and some postulates. In Section 5 we present We will follow the strategy of interpretation via translation, using Montague’s IL as the some examples. Certain rules of the fragment are postponed to Section 6 where they receive intermediate language. But everything we do could also be done by direct interpretation. separate discussion; these are rules that go beyond the simple phrase structure rule schemata of Section 4. (We will do some of this today, and the rest next time: bring this handout next Some abbreviations and notational conventions: α α time too!) We will sometimes write ‘ as a shorthand for TR( ). And sometimes we use the category name in place of a variable over expressions of that category, writing TR(A), or A’, in place of TR(α) when α is an expression of category A. And we will write some of our syntactic rules like simple phrase structure rules. Here is an example of a syntactic rule and corresponding translation rule, and their abbreviations as they will appear below.

MGU052.doc 02/21/05 1:15 AM MGU052.doc 02/21/05 1:15 AM

Formal Semantics, Lecture 2 Formal Semantics, Lecture 2 B. Partee, MGU, February 22, 2005 p.7 B. Partee, MGU, February 22, 2005 p.8

(2) Non-Branching Nodes (NN): If α is a non-branching node, and β is its daughter node, Official Syntactic Rule: If α is an expression of category DET and β is an expression of then [[α ]]œ = [[ β ]]. category CNP, then F0(α,β) is an expression of category NP, where F0(α,β) =  αβ. (3) Functional Application (FA): If α is a branching node, {β,γ} is the set of α’s daughters, Official Semantic Rule: If TR(α) = α' and TR(β) = β', then TR(F0(α,β)) = α'(β'). and [[ β ]] is a function whose domain contains [[ γ ]], then [[ α ]] = [[ β ]] ([[ γ ]]). (4) Predicate Modification (PM) : If α is a branching node, {β, γ} is the set of α’s daughters, → Abbreviated Syntactic Rule: NP DET CNP and [[ β ]] and [[ γ ]] are both in D, then [[ α ]] = λx ∈ De . [[ β ]] (x) = 1 and [[ γ ]] Abbreviated Semantic Rule: NP’ = DET’(CNP’) (x) = 1. (A further principle is needed for intensional functional application, which we will mention 4.2.1. Basic syntactic rules only later.) Basic rules, phrasal: Exactly analogous principles can be written for type-driven translation: S → NP VP NP → DET CNP (1) Terminal Nodes (TN): If α is a terminal node, then TR(A) is specified in the lexicon. CNP → ADJP CNP (2) Non-Branching Nodes (NN): If A → B is a unary rule and A,B are of the same type, then CNP → CNP REL TR(A) = TR(B). VP → TVP NP (3) Functional Application (FA): If A is a branching node, {B,C} is the set of A’s daughters, VP → is ADJP and B’ is of a functional type a → b and C’ is of type a, then TR(A) = TR(B)( TR(C)). VP → is NP (4) Predicate Modification (PM) : If A is a branching node, {B,C} is the set of A’s daughters, Basic rules, non-branching rules introducing lexical categories: and if B’ and C’ are of (same) predicative type a → t, and the syntactic category A can NP → ProperN also correspond to type a → t, then TR(A) = λx[TR(B)(x) & TR(C)(x)]. (i.e. ||A|| = CNP → CN ||B|| ∩ ||C||.) TVP → TV ADJP → ADJ 4.2.2.2. Result of those principles for the translation of the basic rules. VP → IV Function-argument application: S→ NP VP, NP → DET CNP, VP → TVP NP, 4.2.2. Semantic interpretation of the basic rules. VP → is ADJP, VP → is NP, and those instances of CNP → ADJP CNP in which The basic principle for all semantic interpretation in formal semantics is the principle ADJP is of type (e→t)→(e→t). of compositionality; the meaning of the whole must be a function of the meanings of the parts. In the most “stipulative” case, we write a semantic interpretation rule (translation or Example: Consider the rule S→NP VP. If NP is of type (e→t)→t and VP is of type direct model-theoretic interpretation) for each syntactic formation rule, as in classical MG. In e→t, then the translation of S will be NP’(VP’) (e.g., Every man walks). If NP is of type e more contemporary approaches, we look for general principles governing the form of the and VP is of type e→t, then the translation of S will be VP’(NP’) (e.g., John walks). rules and their correspondence (possibly mediated by some syntactic level of “Logical Form”.) Here we are using an artificially simple fragment, and we have presented the Predicate modification: CNP → CNP REL, and those instances of CNP → ADJP CNP in syntactic rules in a form which is explicit but not particularly general; but we have the tools which ADJP is of type e→t. to illustrate a few basic generalizations concerning syntax-semantics correspondence. → → → → 4.2.2.1.Type-driven translation. (Partee 1976, Partee and Rooth 1983, Klein and Sag 1985) Non-branching nodes: NP ProperN, CNP CN, TVP TV, ADJP ADJ. To a great extent, possibly completely, we can formulate general principles for the interpretation of the basic syntactic constructions based on the semantic types of the 4.2.3. Rules of Relative clauses, Quantification, Phrasal Negation. See Section 6. constituent parts. 4.2.4. Type multiplicity and type shifting.

So suppose we are given a rule A → B C, and we want to know how to determine A’ as a We noted in Lecture 1 that classical model-theoretic semantics in the Montague function of B’and C’ (equivalently, TR(A) as a function of TR(B) and TR(C); and tradition requires that there be a single semantic type for each syntactic category. But in Fragment 1, several syntactic types have more than one corresponding semantic type. The ultimately, ||A|| as a function of ||B|| and ||C||.) Similarly for non-branching rules A → B. possibility of type multiplicity and type shifting has been increasingly recognized in the last

decade or so, and there are a variety of formal approaches that accommodate type multiplicity General principles: function-argument application, predicate conjunction, identity. without giving up compositionality. We will not go into details about formal issues here, but The following versions of general type-driven interpretation principles are taken from Heim we do want to include a of categories with multiple semantic types; several were and Kratzer (1995). introduced in Fragment 1, and more will be introduced in later lectures. In the original they are written for direct model-theoretic interpretation.

(1) Terminal Nodes (TN): If α is a terminal node, then [[α]] is specified in the lexicon. MGU052.doc 02/21/05 1:15 AM MGU052.doc 02/21/05 1:15 AM

Formal Semantics, Lecture 2 Formal Semantics, Lecture 2 B. Partee, MGU, February 22, 2005 p.9 B. Partee, MGU, February 22, 2005 p.10

Montague tradition: uniform treatment of NP's as generalized quantifiers, type (e → t) → t. Semantics of Lexicon (MG): John λP[P(John)] (the set of all of John’s properties) Open class lexical items (nouns, adjectives, verbs) translated into constants of appropriate a fool λP∃x[fool(x) & P(x)] type (notation: English expressions man, tall translated into IL constants man, tall, etc.). every man λP∀x[man(x) → P(x)] Interpretation of these constants a central task of . A few open class words (e.g. be, entity, former) sometimes treated as part of the "logical vocabulary". Intuitive type multiplicity of NP's (and see Heim 1982, Kamp 1981): Closed class lexical items: some treated like open class items (e.g. most prepositions), others John "referential use": John type e (esp. "logical" words) given explicit interpretations, as illustrated below. a fool "predicative use": fool type e → t Determiners: every man "quantifier use": (above) type (e→t)→t We have three types of NPs and correspondingly three types of DETs. Not all DETs occur in

all types; the is one of the few that does. For DETs that occur in more than one type, we will Resolution: All NP's have meaning of type (e→t)→t; some also have meanings of types e → subscript the “homonyms” with mnemonic subscripts: e for those that combine with a CNP to and/or e t. General principles for predicting (Partee 1987). Predicates may semantically form an e-type NP, pred for those that form predicate nominals, and GQ for those that form → → → 1 take arguments of type e, e t, or (e t) t, among others. -type NPs. (Note that these are not the types of the DETs themselves, Type choice determined by a combination of factors including by demands of but their own types have unpleasantly long and hard-to-read names.) There are systematic predicates, "try simplest types first" strategy, and default preferences of particular relations among these “homonyms” (Partee 1987), but we are not discussing them here. determiners. → Note the effects of this type multiplicity on type-driven translation. The S NP VP (i) e-forming DETs. rule, for instance, will have two different translations. The VP, we have assumed, is always of ι type e → t. If the NP is of type e, the translation will be VP’(NP’), whereas if the NP is of For the translation of thee, we need to add the iota-operator to IL. → → type (e t) t, the translation will be NP’(VP’), as noted above in Section 4.2.2.2. [See Syntax: If ϕ ∈ MEt and u is a variable of type e, then ιu[ϕ] ∈ MEe. Homework problem #3.] Semantics: ||ιu[ϕ]|| M,w, g = d iff there is one and only one d ∈ D such that ||ϕ|| M,w,g[d/u] =1. M,w, g 4.3. Lexicon. ||ιu[ϕ]|| is undefined otherwise. Here we illustrate the treatment of the lexicon in Montague (1973) (“PTQ”). Montague, not So ιx(king(x)) denotes the unique individual who is king, if there is one, and is undefined if unreasonably, saw a great difference between the study of the principles of compositional there is either no king or more than one king. semantics, which are very similar to the principles of compositional semantics for logical languages as studied in logic and , and the study of lexical semantics, which he The type for determiners as functors forming e-type ("referential") terms is (e → t) → e; the perceived as much more “empirical”. For Montague, it was important to figure out the only determiner of this type we will introduce is thee. For ease of reading, we will give the difference in logical type between easy and eager, or between seem and try, but he did not try translations of representative NPs, rather than for the DET itself; the DET translation can be to say anything about the difference in meaning between two elements with the same formed in each case by λ-abstraction on the CNP (see below, TR(apred)). “structural” or type-theoretic behavior, such as easy and difficult or run and walk. For TR(thee king) = ιx(king(x)) Montague, most lexical items were considered atomic expressions of a given type, and simply translated into constants of IL of the given type. (ii) predicate-forming DETs. DETs as functors forming predicate nominals are of type (e → t) → (e → t). First we simply list some lexical items of various syntactic categories; aside from the TR(apred man) = man category DET, these are all open classes. Then we discuss their semantics. TR(thepred man) = λx[man(x) & ∀y[man(y) → y=x)]]

In later lectures we will be concerned with how best to enrich the semantic information We illustrate the translation of the DET itself with the translation of a . associated with the lexicon in ways compatible with a compositional semantics. pred TR(apred) = λP[P]

ProperN: John, Mary, Bill, ... (iii) generalized quantifier-forming DETs. DET: some, a, the, every DETs as functors forming generalized quantifiers are of type (e→t)→((e→t)→t) ADJ: carnivorous, happy, skillful, tall, former, alleged, old, ... TR(aGQ man) = λP∃x[man(x) & P(x)] CN: man, king, violinist, surgeon, fish, senator, ... ∀ → TV: sees, loves, catches, eats, ... TR(everyGQ man) = λP x[man(x) P(x)] (see Homework problem 4) ∃ ∀ → IV: walks, talks, runs, ... TR(theGQ man) = λP x[man(x) & y[man(y) y=x)]& P(x)]

The copula be: 1 (More on type-shifting in RGGU 2005 Lecture 8 (or see RGGU 2004 Lecture 6 on my website), and more on TR(is) = λPλx[P(x)] ("Predicate!") type multiplicity of adjectives in MGU Lecture 4.) MGU052.doc 02/21/05 1:15 AM MGU052.doc 02/21/05 1:15 AM

Formal Semantics, Lecture 2 Formal Semantics, Lecture 2 B. Partee, MGU, February 22, 2005 p.11 B. Partee, MGU, February 22, 2005 p.12

The types for CN, CNP, and REL are all e → t; so the principle for combining CNP and REL Results (see also Section 5, and Homework problems 7, 8): gives: λy[CNP’(y) & REL’(y)] (Predicate conjunction) TR(is green) = green TR(is apred man) = man The relative clause itself is a predicate formed by λ-abstraction on the variable corresponding TR(is thepred king) = λx[king(x) & ∀y[king(y) → y=x)]] to the WH-word. (Partee 1976 suggests a general principle that all “unbounded movement rules” are interpreted as involving variable-; and λ-abstraction can be taken as the 5. Examples most basic variable-binding operation.) (1) is happy TR(is) = λPλx[P(x)] A syntactically very crude and informal version of the relative clause rule, with its semantic TR(happy) = happy interpretation, can be stated as follows: TR(is happy) = λPλx[P(x)] (happy) ϕ ϕ = λx[happy(x)] = happy Rel Clause Rule, syntax: If is an S and contains an indexed pronoun hei / himi in relativizable position, then the result of adjoining who(m) to S and leaving a trace ei in place (2) The violinist is happy (with e-type interpretation of subject) of hei / himi is a REL. ι TR([NP the violinist]) = x[ violinist(x)] type: e Rel Clause Rule, semantics: If ϕ translates as ϕ’, then REL translates as λxi[ϕ’]. TR([VP is happy]) = happy type: e → t TR([S the violinist is happy] = happy(ιx[violinist(x)]) type: t Semantic derivation corresponding to the syntactic derivation above; compositional (VP meaning applies to NP meaning) translation into IL: (read bottom-to-top) (and see Homework problem 5a)

(3) Every violinist is happy (with GQ-type subject) λy[man(y) & λx [love (Mary, x )] (y)] ∀ → → → 3 3 TR(every violinist) = λP x[violinist (x) P(x)] type (e t) t 3 → TR(is happy) = happy type e t man λ x [love (Mary, x )] ∀ → 3 3 TR(every violinist is happy) = λP x[violinist (x) P(x)](happy) | ∀ → = x[violinist (x) happy(x)] love (Mary, x3)

(4) Every surgeon is a skillful violinist By λ-conversion, the top line is equivalent to: λy[man(y) & love (Mary, y)] (The type of every surgeon must be (e → t) → t; the type of a skillful violinist must be → → → → e t. Assume the type of skillful is (e t) (e t). ) 6.2. Quantifying In. ∀ → TR(every surgeon) = λP x[surgeon (x) P(x)] This is (an informal statement of) Montague’s Quantifying In rule; it is similar to the TR(skillful violinist) = skillful(violinist) Quantifier-Lowering rule of Generative Semantics and Quantifier Raising (QR) of May TR(is a skillful violinist) = TR (a skillful violinist)= TR(skillful violinist) (1977); various alternative treatments of quantifier scope ambiguity exist, including Cooper- ∀ → TR(every surgeon is a skillful violinist) = x[surgeon(x) skillful(violinist)(x)] storage (Cooper 1975) and Herman Hendriks’s flexible typing approach (Hendriks 1988, 1993). 6. Rules of Relative clause formation, Quantifying In, Phrasal Negation. Quantifying In Rule, Syntax: (informally stated): An NP combines with a sentence with respect to a choice of variable (“hei” in MG). Substitute the NP for the first occurrence of the 6.1. (Restrictive) Relative clause formation. variable; change any further occurrences of the variable into pronouns of the appropriate We begin with an illustration of what the rule does before stating it (in a sketchy form). number and gender. Consider the CNP man who Mary loves: Semantic rule: NP’(λxi [S’] ) (The set of properties denoted by the NP includes the Syntactic derivation (very sketchy): property denoted by the λ-expression derived from the sentence.)

CNP We illustrate with two derivations for the ambiguous sentence Every student read a book. 3 CNP REL: who Mary loves [e3] Syntactic derivation (i) (rough sketch; read from bottom to top. Bold is used here to show | | which variables are substituted for at each step.) CN S: Mary loves him3 | man

MGU052.doc 02/21/05 1:15 AM MGU052.doc 02/21/05 1:15 AM

Formal Semantics, Lecture 2 Formal Semantics, Lecture 2 B. Partee, MGU, February 22, 2005 p.13 B. Partee, MGU, February 22, 2005 p.14

S: every student read a book 3 Syntactic rules for conjunction: Corresponding semantic rules: NP: every student S: he3 read a book S → S and S S’ = S1’ & S2’ 3 S → S or S S’ = S1’ ∨ S2’ NP: a book S: he3 read him2 VP → VP and VP VP’ = λx [VP ’(x)& VP ’(x)] 1 2 VP → VP or VP VP’ = λ x [VP ’(x) ∨ VP ’(x)] Compositional Translation: (every student)’(λx [(a book)’(λx [read (x , x )] )] ) 1 2 3 2 3 2 NP → NP and NP NP’ = λP[NP ’(P) & NP ’(P)] 1 2 → λ ∨ Rough paraphrase: Every student has the property that there is a book that he read. NP NP or NP NP’ = P [NP 1’(P) NP 2’(P)]

If you write out the interpretations of the NPs and apply Lambda-Conversion as many times The NP conjunction and disjunction rules presuppose that the NPs are interpreted as as possible, the result will be (some alphabetic variant of) the first-order PC formula generalized quantifiers, type <,t>; P is a variable of type . (Conjoined NPs of type ∀x(student(x) → ∃y(book(y) & read(x,y))). e can be interpreted as groups, but not as conjoined by Boolean conjunction.)

Syntactic derivation (ii) Examples: Some animals swim and some animals fly. (S-conjunction) S: every student read a book Some animals swim and fly. (VP-conjunction) 3 Every fish and some birds swim. (NP-conjunction) NP: a book S: every student read him2 Every painting and every statue was photographed or (was) videotaped. (NP-conjunction and 3 VP-conjunction (or conjunction of participles, if we omit the second ‘was’, but it’s equivalent NP: every student S: he3 read him2 to VP conjunction). The rules do correctly “predict” which conjunction has wider scope. (Optional unlisted extra homework question: work out the last example. Treat “was Compositional Translation: (a book)’(λx2[(every student)’(λx3 [read (x3 , x2 )] )] ) [See photographed” and “was videotaped” as simple 1-place predicates for this exercise.) Homework problem 6.] Paraphrase: Some book has the property that every student read it. We could extend the rules above, and generalize them (as is done in Partee and Rooth 1983), After applying Lambda-Conversion as many times as possible, the result will be (some so as to include further types of phrasal conjunction such as the following: alphabetic variant of) the first-order PC formula ∃y(book(y) & ∀x(student(x)→ read(x,y))). John bought and read a new book. (TV conjunction) No number is even and odd. (Predicate ADJP conjunction) Observation: Compositional semantics requires that every ambiguous sentence be Mary saw an old and interesting manuscript. (Pre-nominal ADJP conjunction.) explainable on the basis of ambiguous lexical items and/or multiple syntactic derivations. Semantic structure mirrors syntactic part-whole structure, which in Montague Grammar is In fact, we do not have to “stipulate” the rules one-by-one as we have done above; it is represented by syntactic derivational structure, not surface structure. There are different possible to predict them in a general way from the types of the expressions being conjoined. theories of the semantically relevant syntactic structure: “Derivation trees” or “analysis But that goes beyond the scope of these lectures; see Partee and Rooth 1983. trees” (MG), LF (Chomskian GB or Minimalist theory), Tectogrammatic Dependency Trees (Prague), Deep Syntactic Structure (Mel’čuk) Underlying Structure (Generative Semantics), 6.4. Phrasal and lexical negation. ... . GPSG, HPSG, and various contemporary versions of are attempts to As an additional augmentation of our grammar which adds further illustration of the represent all the necessary syntactic information directly in a single “level” of syntax. application of the lambda calculus, let us consider the relations among sentence negation, phrasal negation, and lexical (prefixal) negation. 6.3. Conjunction. The syntax of sentential negation in English is slightly complicated because of its One simple and elegant application of lambda abstraction which Montague used in PTQ is its interaction with the system of verbal auxiliaries, which we have not included in our simple use in defining the interpretation of “Boolean” phrasal conjunction, disjunction, and negation grammar. Let us ignore the syntactic complexities here and work with a “Logical Form” in terms of the corresponding sentential operations. grammar in which we have a simple phrase structure rule S NEG S. And let us add to the “Boolean” phrasal conjunction, illustrated in all the examples below, is distinguished lexicon an “NOT” of syntactic category NEG, of semantic type t t, whose from “part-whole” or “group” conjunction, illustrated by “John and Mary are a happy couple” translation into IL is simply ¬. Then the interpretation of NEG S, by function-argument and “The flag is red and white”, which are not equivalent, respectively, to “John is a happy application, will just be ¬S’. couple and Mary is a happy couple” and “The flag is red and the flag is white”. To illustrate this application, we add a few syntactic and semantic rules to our Now what about phrasal negation, like not every boy, not today, Mary but not John, not very intelligent, not love Bill? Negation, like conjunction, is a “Boolean” operation, and it fragment. Note: in the semantic rules, we use S1 and S2, etc., to refer to the first and second S in the syntactic rule. MGU052.doc 02/21/05 1:15 AM MGU052.doc 02/21/05 1:15 AM

Formal Semantics, Lecture 2 Formal Semantics, Lecture 2 B. Partee, MGU, February 22, 2005 p.15 B. Partee, MGU, February 22, 2005 p.16 is easy to define its interpretation with expressions of various types on the basis of its types of individuals, truth-values, and n-ary relations over individuals, IL has a rich system of interpretation with sentences. For example: types which makes it much easier to achieve a (relatively) close fit between expressions of Syntax: VP → NEGVP VP various categories of a natural language and expressions of IL. The types serve as syntactic Semantics: Type: Since the type of VP is e→t, the type of NEGVP must be (e → t)→(e → t). categories for the expressions of IL; because of the role of IL as an intermediate language in TR(VP2) = TR( NEGVP)(TR(VP1) the semantic interpretation of natural language, the same types are referred to as semantic = λP[λx[¬ P(x)]](TR(VP1)) types for expressions of natural language. = λx[¬VP1' (x)] The types of Montague’s IL are as follows: Basic types: e (entities), t (truth values) Similar rules for negation of other phrasal categories can be derived in a uniform way. Functional types: If a,b are types, then is a type (the type of functions from a-type (See Homework Problem 5b, which asks for NEGNP and optionally NEGDET.) things to b-type things.) Note: We use interchangeably the two notations and a 6 b, Both negation and conjunction (and disjunction) thus have natural extensions to a both of which are common in the literature. wide range of types (see Partee and Rooth 1983), with meanings systematically derivable Intensional types: If a is a type, then is a type (the type of functions from possible from their basic meanings, which apply to sentences. worlds to things (extensions) of type a.) Lexical negation, as in unhappy, impossible, unbroken, inedible, can also be defined (In some systems, the basic type t is taken as intensional, interpreted as the type of with the use of lambdas: rather than of truth-values. In general, we will mostly ignore intensionality in TR(unhappy) = λx[¬TR(happy)(x)] = λx[¬happy(x)] these lectures, working most of the time with extensional versions of our fragments and

mentioning intensionality only where directly relevant. But that is only for simplicity of exposition; in general, a thoroughly intensional semantics is presupposed.) ======A.2.1.2. Model structures. Appendix: Montague’s intensionsal logic, with lambdas and types. In the first lecture, we introduced the simple model structure M1 for interpreting the A.1 Introduction predicate calculus. A model M for the typed intensional logic IL has much more structure, but that structure is built up recursively from a small set of primitives. Tools like Montague’s Intensional Logic are important in making a more satisfactory compositional analysis of natural language semantics possible. What are the differences Model structure for IL: M = . Each model must contain: between Montague’s IL and PC? Here are some of the most important: A domain D of entities (individuals) (i) The rich type structure of IL. A set W of possible worlds (or -time pairs, or possible situations) (ii) The central role played by function-denoting expressions. All of the types except the ≤ : an ordering (understood as temporal order) on W basic types e and t are functional types, and all of the expressions of IL except those of types I: Interpretation function which assigns semantic values to all constants. e and t are expressions which denote functions. Functions may serve as the arguments and as the values of other functions. In particular, all relations are also represented as functions. The domains of possible for expressions of type a (relative to D,W) are defined (iii) The inclusion of the operation of “functional application” or “function-argument recursively as follows: application”, the application of a function to its argument. De = D (iv) The use of lambda-expressions. The lambda-operator is the basic tool for building Dt = {0,1} expressions which denote functions. D = {f | f: Da → Db }(i.e. the set of all functions f from Da to Db.) (v) In place of the one “world” of PC (where there is in effect no distinction between a D = {f | f: W → Da }(i.e. the set of all functions f from W to Da .) “world” and a model), the models of IL include a set of possible worlds. Possible worlds are The semantic interpretation of IL also makes use of a set G of assignment functions g, crucially connected with / distinction and with intensional types. Possible functions from variables of all types to values in the corresponding domains. worlds, in particular, underlie the interpretation of modal operators and referential opacity. (vi) The models of IL also include, in one way or another, a structure of time, used among Each expression of IL has an intension and, at each w in W, an extension. The intension is other things in the interpretation of tense operators like PAST in the fragment below. relative to M and g; the extension is relative to M, w, and g. But we will not discuss and extensions in this lecture. A.2. Intensional Logic (IL). A.2.2. Atomic expressions (“lexicon”), notation, and interpretation. A.2.1. Types and model structures. The atomic expressions of IL are constants and variables; there are infinitely many constants A.2.1.1. Types and infinitely many variables in each type. Montague introduced a general nomenclature for constants and variables of a given type, using c and v with complex subscripts indicating type Montague’s IL is a typed intensional language; unlike the predicate calculus, which has and an index. In practice, including Montague’s, more mnemonic names are used. Our variables of only one type (the type of entities or individuals), and expressions only of the conventions will be as follows:

MGU052.doc 02/21/05 1:15 AM MGU052.doc 02/21/05 1:15 AM

Formal Semantics, Lecture 2 Formal Semantics, Lecture 2 B. Partee, MGU, February 22, 2005 p.17 B. Partee, MGU, February 22, 2005 p.18

(The next two pairs of rules, concerning the “up” and “down” operators, are crucial Constants of IL will be written in non-italic boldface, and their names will usually reflect the for intensionality, but we will not discuss them and will not use them.) English expressions of which they are translations: man, love, etc. Their types will be ∧ specified. Variables of IL will be written in italic boldface, usually observing the following Syntactic Rule 4: (“up”-operator.) If α ∈ MEa, then [ α] ∈ ME. ∧ conventions as to types: Semantic Rule 4: || [ α]||M, w,g is that function h of type < s,a> such that for any w’ in Type e: w,x,y,z, with and without subscripts or primes (this modifier holds for all types.) W, h(w’) = ||α||M, w’,g. Type : P, Q ∨ Syntactic Rule 5: (“down”-operator.) If α ∈ ME, then [ α]0 MEa. ∨ Various relational types such as >: R Semantic Rule 5: || [ α]||M, w,g is ||α||M, w,g(w) The type of generalized quantifiers: T The next two pairs of rules, function-argument application and lambda-abstraction, are The interpretation of constants is given by the interpretation function I of the model, and the among the most important devices of IL, and we will make repeated use of them. interpretation of variables by an assignment g, as specified in Rule 1 below. Function-argument application: α ∈ β ∈ α β ∈ A.2.3. Syntactic rules and their model-theoretic semantic interpretation Syntactic Rule 6: If ME and MEa, then ( ) MEb. Semantic Rule 6: ||α(β)||M,w,g = ||α||M,w,g (||β||M,w,g) The syntax of IL takes the form of a of the set of “meaningful expressions of type a”, ME , for all types a. The semantics gives an interpretation rule for a Lambda-abstraction: each syntactic rule. α ∈ α ∈ Syntactic Rule 7: If MEa and u is a variable of type b, then λu[ ] ME. α M, w,g → Note: when giving syntactic and semantic rules for IL, as for predicate logic, we use a Semantic Rule 7: || λu[ ] || is that function f of type b a such that for any object d of α M, w,g[d/u] metalanguage which is very similar to IL; but we are not boldfacing the constants and type b, f(d) = || || . variables of the metalanguage. The metalanguage variables over variables are most often ======chosen as u or v. REFERENCES.

Heim, I. and A. Kratzer (1998). Semantics in . Oxford:Blackwell The first rule is a rule for atomic expressions, and the first semantic rule is its interpretation: Klein, Ewan and Ivan Sag (1985), "Type-driven translation", Linguistics and 8, 163-201. Syntactic Rule 1: Every constant and variable of type a is in MEa. Lewis, David (1970) "General semantics" Synthese 22, 18-67; reprinted in D.Davidson and G.Harman M,w,g Semantic Rule 1: (a) If " is a constant, then ||α|| = I(α)(w). (eds.), Semantics of Natural Language. Dordrecht: Reidel (1972), 169-218. Also reprinted in (b) If " is a variable, then ||α||M,w,g = g(α). Partee, Barbara H. ed. 1976. Montague Grammar. New York: Academic Press. Montague, R. (1973) "The Proper Treatment of Quantification in Ordinary English," in K.J.J. Note: The recursive semantic rules give extensions relative to model, world, and assignment. Hintikka, J.M.E. Moravcsik, and P. Suppes, eds., Approaches to Natural Language, Reidel, Read “||α||M,w,g ” as “the semantic value (extension) of alpha relative to M, w, and g.” The Dordrecht, 221-242, reprinted in Montague (1974) 247-270. Also reprinted in Portner, Paul, and interpretation function I assigns to each constant an intension, i.e. a function from possible Partee, Barbara H. eds. 2002. Formal Semantics: The Essential Readings. Oxford: Blackwell Publishers. worlds to extensions; applying that function to a given world w gives the extension. Montague, Richard (1974) Formal Philosophy: Selected Papers of . Edited and with an introduction by Richmond Thomason, New Haven: Yale Univ. Press. Syntactic Rule 2. (logical connectives and operators that apply to formulas, mostly from Partee, Barbara H. (1976) "Semantics and syntax: the search for constraints" in C.Rameh (ed.) propositional and predicate logic, plus some modal and tense operators.) If ϕ,ψ MEt, and u is Georgetown University Round Table on Languages and Linguistics 1976, Georgetown: a variable of any type, then ¬ϕ, ϕ&ψ, ϕ∨ψ, ϕ→ψ, ϕ↔ψ (also written as ϕ≡ψ), ∃uϕ, ∀uϕ, Georgetown University School of Languages and Linguistics (99-110). ϕ, PASTϕ ∈ MEt. Note: “ ϕ” is read as “Necessarily phi”. Partee, Barbara (1987) "Noun phrase interpretation and type-shifting principles", in J. Groenendijk, D. de Jongh, and M. Stokhof, eds., Studies in Representation Theory and the Theory of Semantic Rule 2: Generalized Quantifiers, GRASS 8, Foris, Dordrecht, 115-143. Reprinted in: Portner, Paul, and ϕ ϕ ϕ∨ ϕ→ ϕ↔ ∃ ϕ ∀ ϕ Partee, Barbara H. eds. 2002. Formal Semantics: The Essential Readings. Oxford: Blackwell (a) ¬ , &ψ, ψ, ψ, ψ, u , u as in predicate logic. Publishers. ϕ M,w,g ϕ M,w’,g (b) || || = 1 iff ||| || = 1 for all w’ in W. Partee, Barbara and Mats Rooth, "Generalized conjunction and type ambiguity", in R. Bauerle, C. (c) || PAST ϕ||M,w,g = 1 iff ||ϕ||M,w’,g = 1 for some w’ ≤ w. (This is a simplification; here we are Schwarze, and A. von Stechow (eds.) Meaning, Use and Interpretation of Language, Walter de treating each w as a combined “world/time index”, possibly a situation index; w’ ≤ w if w’ is Gruyter, Berlin (1983) 361-383. Reprinted in: Portner, Paul, and Partee, Barbara H. eds. 2002. a temporally earlier slice of the same world as w.) Formal Semantics: The Essential Readings. Oxford: Blackwell Publishers. Partee, B., A. ter Meulen, and R.E. Wall (1990) Mathematical Methods in Linguistics, Dordrecht: Kluwer Academic Publishers. Syntactic Rule 3: (=): If α,β ∈ MEa, then α=β ∈ MEt. α β M,w,g α M,w,g β M,w,g Portner, Paul, and Partee, Barbara H. eds. 2002. Formal Semantics: The Essential Readings. Oxford: Semantic Rule 3: || = || = 1 iff || || = || || . Blackwell Publishers.

MGU052.doc 02/21/05 1:15 AM MGU052.doc 02/21/05 1:15 AM

Formal Semantics, Lecture 2 Formal Semantics, Lecture 2 B. Partee, MGU, February 22, 2005 p.19 B. Partee, MGU, February 22, 2005 p.20

HOMEWORK #1, Due March 8. (i) S

Do at least problems 1 and 2. 2 pages should be enough; 4 pages maximum. There is an extra 3 page of “homework help” for this homework. First try to do it without looking at the help, NEG S then look at the help, and then try it again if you did it wrong the first time. Bring questions to 3 seminar next week if you would like additional help then. Optional extra problems, if you are NP VP able to do more: 3-8. Note: Don’t forget to give me Homework #0, the “Anketa”! Then figure out what the type and translation should be for a not which can apply to NPs of ======type (e → t) → t, and work out the translation for the sentence under an NP-negation analysis, where the syntactic structure begins as follows: 1. (a) Write down the translation into the λ-calculus of “A student walks and talks”. This (ii) S handout already shows the translations of “a student” and “walks and talks”. Put them 3 together by “function-argument application”. NP VP (b) Apply λ-conversion to simplify the formula. There will be two applications, and the 3 resulting formula should have no λ‘s. NEG NP (c) Write down the translation of “A student walks and a student talks”; simplify by λ- conversion. There is a third possibility, which is to apply not to every; if you have figured out how to do (d) The two formulas (if you did parts (a-c) correctly) are not equivalent. Describe a (i) and (ii), you’ll be able to figure out how to do the third; it’s just more lambdas. Don’t do it situation (a model) in which one of them is true and the other one is false. unless you really want to. What might be more interesting would be to work on linguistic arguments to try to decide how many of the three are real possibilities for English, and/or the 2. In the predicate calculus, the sentence “No student talks” can be represented as follows: same question for the corresponding Russian sentence. ¬∃x [student(x) & talk(x)] or equivalently as ∀x[student(x) → ¬talk(x)] But in the predicate calculus, there is no way to represent the meaning of the NP “no 6. Fill in the missing steps in the derivation of reduced forms of translations of the example student”. Using the λ-calculus in the way illustrated above for the NPs “every student”, “a every student reads a book, on derivation (ii) in section 6.2. Use Montague’s generalized student”, “the king”, write down a translation for the NP “no student”. (There are two quantifier interpretations of every student and a book, as given in Section 3, repeated under logically equivalent correct answers; write down either or both.) “generalized quantifier-forming DETS” in Section 4.3. This is an exercise in compositional interpretation and lambda-conversion. Additional questions, optional. 7. Fill in all the steps to show why TR(is apred man) = man. (End of section 4.3) 3. Write out the semantic derivation of John walks two ways, once using Montague’s generalized quantifier interpretation of John, once using the type-e interpretation of John. 8. Write the translations of the in each of its three types. (Section 4.3)

4. Write the translation for every, by abstracting on the CNP in the given translation of every man. Hint: the translation of every, like that of every DET, should begin with λQ, where Q is a When you do homework, feel free to write questions on your homework paper. variable of type e → t, the type of the CNP with which the DET “wants to” combine. ======

5. Work out the translations, using lambda-conversion for simplification of results, of the following. Always apply lambda-conversion as soon as it is applicable, so that the formulas do not become more complex than necessary. (a) Every violinist who loves Prokofiev is happy. (b) Not every violinist is unhappy.

Note on exercise 5(b). (Not every violinist is unhappy.) Note: Work this out two ways, which should come out equivalent. First pretend that the not is sentential negation, although according to the rules of English syntax, this is not a possible position for a sentential not. So the first syntactic structure should begin as follows:

MGU052.doc 02/21/05 1:15 AM MGU052.doc 02/21/05 1:15 AM

Formal Semantics, Lecture 2 B. Partee, MGU, February 22, 2005 p.21

HELP FOR HOMEWORK #1 Help with Problem 1. Instead of giving an answer, here is an answer to a similar problem, which we’ll call Problem 1*.

Problem 1*. (a) Write down the translation into the λ-calculus of “Every student walks and talks”. The handout already shows the translations of every student and walks and talks. Just put them together by function-argument application. (b) Apply λ-conversion to simplify the formula. There will be two applications, and the resulting formula should have no λ‘s. (c) Write down the translation of “Every student walks and every student talks”; simplify by λ-conversion. (d) Are the two formulas equivalent? Give an argument.

Answer. (1*) (a) every student : λP[∀x ( student(x) → P(x) )] walks and talks: λy[(walk (y) & talk(y))] every student walks and talks: λP[∀x (student(x) → P(x))] (λy[(walk (y) & talk(y))])

(b) Simplify the expression by two applications of λ-conversion. Step 1: λP[∀x (student(x) → P(x))] (λy[(Walk (y) & Talk(y))]) ≡ ∀x (student(x) → λy[(Walk (y) & Talk(y))] (x)) Step 2: ∀x (student(x) → λy[(walk (y) & talk(y))] (x)) ≡ ∀x (student(x) → (walk (x) & talk(x)))

(c) Every student walks and every student talks: λP[∀x ( student(x) → P(x) )] (walk) & λP[∀x ( student(x) → P(x) )] (talk) ≡ ∀x ( student(x) → walk(x)) & ∀x ( student(x) → talk(x) ) It doesn’t matter whether the same variable x is used in both formulas or not; this is equivalent to: ∀x ( student(x) → walk(x)) & ∀y ( student(y) → talk(y) )

(d) Use first-order predicate logic to argue that the last formula in (b) and the last formulas in (c) are equivalent. Both formulas require that every student have both properties.

Note: If two formulas ARE equivalent, you should use what you know about predicate logic to try to prove the equivalence (either or or an informal argument). If two formulas are NOT equivalent, then you should construct a model (often a very small model is enough) in which one of the formulas is true and the other one is false: that’s always the best and simplest way to show non-equivalence.

MGU052.doc 02/21/05 1:15 AM